• The topic of this special issue is “Evolutionary or plastic responses to climate change”. The underlying question – whether observed responses have a genetic basis or are expressions of phenotypic plasticity – is an empirical one, to which evolutionary theory can only contribute indirectly.
• In particular, current theory is unable to make quantitative predictions about the magnitude of plastic responses. Furthermore, the challenges discussed in the introductory paper by Merilä and Hendry do not apply to
• Much of the relevant theory has not been developed with climate change specifically in mind (a notable exception being models of “evolutionary rescue”).

In which ways can evolutionary theory contribute to the question at hand?

• Identify factors that influence the response to environmental change.
• Make qualitative predictions about the effect of different environmental and genetic variables.
• Provide rough rules of thumb for the magnitude of genetic (but not plastic, wee above) changes.
• Make suggestions about measurements and scaling (e.g. Hereford et al. 2004, Hansen and Houle 2008, Gingerich 2009)

Our review will have the following structure

• Genetic responses (focus on expected rates and evolutionary rescue)
  • Univariate case
  • Multivariate case
• Plastic responses

Consider now the case of gradual shift of a one-dimensional phenotypic optimum. Typically, a Gaussian fitness function is assumed:

<latex> w(z) = \exp\left(-\frac{(z-\theta_t)^2}{2\sigma_s^2}\right) </latex>

Here, z is the phenotype of an individual, theta_t is the optimum at time t, and the “variance” sigma^2 measures the width of the fitness function (i.e., selection is strong sigma^2 is small). Here and later on, our notation mainly follows … (Bürger and Lynch 1995?). Frequently, it is assumed that, at time 0, z = theta_0, that is, the population is perfectly adapted before the optimum starts moving. Movement of the optimum is typically assumed to be linear, potentially accompanied by stochastic white noise:

<latex> \theta_t = kt + \epsilon_t </latex>

where <latex>\epsilon_t</latex> is drawn from a normal distribution with mean 0 and variance <latex>\sigma_\theta^2</latex>. Setting <latex>\sigma_\theta^2 = 0</latex> recovers the deterministic case. This basic model has first(?) been analyzed by Lynch and Lande (1993) using a quantitative genetic framework, and later been extended by Bürger and Lynch (1995), … Since its behavior is highly instructive, we will describe it in some detail.

Assume that the trait z has a polygenic basis. In the simplest case (additive genetics, no plasticity), its variance can be decomposed into <latex>\sigma_p^2 = \sigma_g^2 + \sigma_e^2</latex>, where <latex>\sigma_p^2</latex> is the phenotypic variance, <latex>\sigma_g^2</latex> is the additive genetic variance, and <latex>\sigma_e^2</latex> is the environmental variance (variation due to developmental instability or micro-environmental fluctuations). Assuming non-overlapping generations, evolution of the mean phenotype can be described by

<latex> \Delta \bar z = \sigma_g^2 \beta </latex>

where the selection gradient <latex>\beta</latex> is given by

<latex> \beta = \frac{d \ln \bar w}{d \bar z}. </latex>

<latex>\beta</latex> measures the proportional change in mean fitness per unit change of the mean phenotype. With the Gaussian fitness function given above

<latex> \beta = \frac{\bar z - \theta_t}{\sigma_s^2 + \sigma_e^2}, </latex>

that is, the selection gradient increases linearly over time. As the optimum starts moving, an initially well-adapted population (z = \theta_0 = 0) will start to evolve, but initially, the selection gradient is small and, hence, the rate of adaptation is smaller than the rate of environmental change and the population will gradually slip off the optimum. However, as the distance to the optimum increases, so does the selection gradient, until finally a state of dynamic equilibrium is reached at which the rate of evolution exactly matches the rate of environmental change. At this equilibrium

<latex> \Delta \bar z^* = … </latex>

Even if the genetic variance is assumed to be constant, whether or not the population can actually follow the optimum depends on the population growth rate (mean fitness) at the equilibrium. The mean fitness can be decomposed into

<latex> \bar w_t = w_{max} - l_g - l_d </latex>